Department: PhysicsModule co-ordinator: Dr. Andrew HigginbothamCredit value: 20 creditsCredit level: IAcademic year of delivery: 2022-23

- See module specification for other years: 2021-22

## Pre-requisite modules

## Co-requisite modules

## Prohibited combinations

Occurrence | Teaching period |
---|---|

A | Autumn Term 2022-23 to Spring Term 2022-23 |

Thermodynamics is a branch of physics that can be applied to any system in which thermal processes are important, although we will concentrate on systems in thermal equilibrium. It is based on four laws (derived from experimental observation) and makes no assumptions about the microscopic character of the system. It is therefore very powerful and general. We will introduce these laws, consider their consequences and apply them to some simple systems.

This part of the module will prepare you for applications in different branches of physics, including solid state physics, and provide a foundation for the model-dependent statistical mechanics approach.

The Solid State Physics part of the module will introduce a key application of the concepts of thermodynamics in understanding the properties of crystalline solids. The main aims here are -

- the understanding of the structure of crystalline solids, including how it is experimentally determined, and that real materials exhibit departures from ideal crystallinity
- the role of lattice vibrations and phonons in the electrical and thermal properties of materials
- the development and detailed description of classical free electron theory to describe the electrical and thermal behaviour of metals

*Thermodynamics*

- Define and explain fundamental concepts such as system, state function, quasistatic reversible process, thermodynamic equilibrium and equation of state.

- State the Zeroth Law of Thermodynamics; explain how this leads to the definition of empirical temperature, describe how the International Temperature Scale is realised and perform calculations related to empirical temperature scales.

- State the First Law of Thermodynamics and show how this leads to a definition of the internal energy, U, as a state function and to the conservation law dU = dW + dQ.

- Define bulk parameters, such as the principal heat capacities, and perform calculations requiring application of the First Law.

- Explain the concept of an ideal reversible heat engine, describe a Carnot cycle and derive the efficiency of a Carnot engine.

- State the Kelvin-Planck and Clausius forms of the Second Law of Thermodynamics and show they are equivalent. Use this law to prove Carnot’s theorem and its corollary.

- Show how thermodynamic temperature may be defined from the Second Law. Perform calculations relating to ideal engines, refrigerators and heat pumps.

- Derive Clausius’ theorem from the Second Law and show how this theorem leads to the definition of entropy, S. Prove that S is a state function. Derive the entropy form of the First Law. Calculate entropy changes for simple irreversible processes.

- Define the Helmholtz and Gibbs functions and show how these are related to conditions of thermodynamic equilibrium.

- Derive the four Maxwell relations for systems with two degrees of freedom and use them in calculations and derivations.

- Define the order of a phase transition in terms of derivatives of the Gibbs function.

- Derive the Clausius-Clapeyron equation for a first order phase transition and apply it to solid-liquid, liquid-vapour and solid-vapour phase transitions. Obtain Ehrenfest’s equations for second order transition.

- State the Third Law of Thermodynamics and describe some of the consequences for the behaviour of systems at low temperatures.

- Discuss and use (in quantitative calculations) the fundamental ideas of thermodynamics in a range of systems such as (i) showing that U is independent of T for an ideal gas; (ii) deriving the TdS equations and use them to describe the behaviour of the principal specific heat capacities; (iii) applying a thermodynamic approach to the elastic deformation of a rod; (iv) deriving the equations for the Joule and Joule-Kelvin coefficients and explaining how the Joule-Kelvin effect is used in the liquefaction of gases; (v) the thermodynamic analysis of black body radiation etc.

*Solid State*

- Describe the structure of crystalline materials in terms of lattice and basis, and describe structural elements such as directions and planes using standard notations

- Understand the origins, nature and consequences of defects within otherwise ideal materials

- Understand the concept of reciprocal space and its role in describing and quantifying wave phenomena in solids

- Derive the conditions for x-rays to diffract from solids, including the concept of the structure factor

- Derive dispersion relations for vibrations in solids, and describe their interpretation in terms of both normal modes and phonons

- Understand how density of states and occupation can be used to calculate macroscopic properties of solids

- Describe the origins of the classical (Dulong-Petit) law of heat capacity, and discuss its failure at low temperature

- Understand the role of quantisation in describing low temperature lattice heat capacities, and discuss the Einstein and Debye models of heat capacity

- Explain the origins of thermal conductivity and thermal expansion of the lattice

- Derive results for electrical conduction, thermal conduction and heat capacity of a classical free electron gas, and describe its relevance to metallic systems

- Explain how application on quantum theory can resolve shortcomings in the classical model of free electron gasses

Describe the successes and failures of a classical approach to free electron theory, including the positive sign of the Hall coefficient in some metals

In addition to co-requisites listed above, students should either take PHY00032I or PHY00036I alongside Thermodynamics and Solid State I

**Syllabus ****–**

**Thermodynamics (Dr Pratt)**

- Introduction to systems, state functions, quasistatic reversible processes and equations of state.

- The Zeroth Law of Thermodynamics, empirical temperature scales and thermometers. The International Temperature Scale.

- The First Law of Thermodynamics and internal energy U. dU = dQ + dW. Perfect and imperfect differentials. Discussion of whether U, Q and W are state functions.

- Definition of heat capacity in general terms and expressions for C
_{P}and C_{V}for a compressible fluid. Description of how U varies with P and V for ideal and real gases. Proof that C_{P}- C_{V}= nR for an ideal gas. Quasistatic adiabatic process for an ideal gas. The van der Waals and virial equations for gases and how they attempt to account for the behaviour of real gases.

- Definition of enthalpy and identification of changes in specific enthalpy with specific latent heat and heat of reaction. Proof that C
_{P}= ( H/ T)_{P}. How the work done in a continuous flow process is related to changes in enthalpy and application of this result to ideal continuous flow processes.

- Ideal reversible heat engines. The Carnot cycle and derivation of an expression for the efficiency of an engine operating in a Carnot cycle.

- The Kelvin-Planck and Clausius forms of the Second Law of Thermodynamics and demonstration of their equivalence. Carnot's theorem and its corollary. Definition of thermodynamic temperature from the second law.

- Figures of merit of ideal refrigerators and heat pumps. The Otto and Diesel cycles.

- Clausius' theorem. Entropy S. dU = TdS – PdV. Changes in entropy for some simple irreversible processes.

- Helmholtz and Gibbs functions. Relationship to conditions of thermodynamic equilibrium.

- Maxwell relations for systems with two degrees of freedom.

- The Third Law of Thermodynamics and consequences for behaviour of systems at low temperature.

- Various applications of the fundamental ideas of thermodynamics including at least some of the following: (i) showing that the internal energy of an ideal gas is independent of p and V; (ii) derivation of the two 'TdS Equations' and use of them to describe the behaviour of the principal specific heat capacities; (iii) a thermodynamic approach to the elastic deformation of a rod; (iv) the application of thermodynamics to black body radiation; (v) derivation of the equations for the Joule and Joule-Kelvin coefficients and explanation of how the Joule-Kelvin effect is used in the liquefaction of gases.

- Definition of the order of a phase transition in terms of the derivatives of the Gibbs function.

- Derivation of the Clausius-Clapeyron equation for a first order phase transition and application of this equation to solid-liquid, liquid-vapour and solid-vapour phase transitions.

- Ehrenfest's equation for a second order phase transition.

**Solid State (Dr Higginbotham)**

- The concepts of point and translational symmetry
- The definition of crystal structures in terms of lattice and basis

- The use of Miller indices to index crystal planes in structures.

- The use of Miller indices to indicate direction and inter-planar spacing in a cubic crystals and derivation of expressions to do so.

- The Miller-Bravais system for indexing of hexagonal systems.

- Point Defects (vacancies, interstitials and impurities). Dislocations and Burgers vector. Stacking and planar defects (stacking faults and twins)

- The reciprocal lattice and Brillouin Zones, including the Wigner-Seitz construction. Extended, repeated and reduced zone schemes.

- Derivation of von Laue's approach for X-ray diffraction by crystals.

- Derivation and use of Bragg’s Law and the Ewald sphere.

- The structure factor and its relation to the reciprocal lattice.

- Use of the structure factor to determine crystal structure in a diffraction experiment.

- Lattice vibrations: the mathematical description of a vibrational wave for planes of atoms containing 1 or 2 atoms per unit cell and the derivation of the dispersion relation between and k, optical and longitudinal modes of vibration

- The concept of density of states and occupation. Their use in determining total and mean energies of a system.

- The breakdown of the classical Dulong-Petit Law for the specific heat capacity of a solid and introduction to the ideas of the Debye and Einstein models including the Debye temperature.

- Thermal conduction and expansion in a solid including the phonon contribution to the mean free path.

- Classical free electron theory (The Drude model) for the electrical and thermal properties of metals, and its limitations.

- Derivation of classical expressions for electrical conductivity, thermal conductivity, the electronic contribution to specific heat capacity, mean free path and the Wiedemann-Franz Law.

- Matthiessen’s Rule for the resistivity of metals.

- Hall effect and the sign of the Hall coefficient.

- Lattice vibrations: the mathematical description of a vibrational wave for planes of atoms containing 1 or 2 atoms per unit cell and the derivation of the dispersion relation between and k, optical and longitudinal modes of vibration

- The concept of density of states and occupation. Their use in determining total and mean energies of a system.

- The breakdown of the classical Dulong-Petit Law for the specific heat capacity of a solid and introduction to the ideas of the Debye and Einstein models including the Debye temperature.

- Thermal conduction and expansion in a solid including the phonon contribution to the mean free path.

- Classical free electron theory (The Drude model) for the electrical and thermal properties of metals, and its limitations.

- Derivation of classical expressions for electrical conductivity, thermal conductivity, the electronic contribution to specific heat capacity, mean free path and the Wiedemann-Franz Law.
- Matthiessen’s Rule for the resistivity of metals.
- The concept of a quantum electron gas and its application to metals
- Hall effect and the sign of the Hall coefficient.

Task | Length | % of module mark |
---|---|---|

Closed/in-person Exam (Centrally scheduled)Solid State Physics I Exam |
1.5 hours | 40 |

Closed/in-person Exam (Centrally scheduled)Thermodynamics Exam |
1.5 hours | 40 |

Essay/courseworkSolid State Physics I Assignment |
N/A | 10 |

Essay/courseworkThermodynamics Assignment |
N/A | 10 |

None

Task | Length | % of module mark |
---|---|---|

Closed/in-person Exam (Centrally scheduled)Solid State Physics I Exam |
1.5 hours | 50 |

Closed/in-person Exam (Centrally scheduled)Thermodynamics Exam |
1.5 hours | 50 |

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

*Thermodynamics*

Adkins CJ: *Equilibrium thermodynamics* (CUP)***

Zemansky MW and Dittman RH: *Heat and thermodynamics* (McGraw-Hill)***

Blundell SJ and KM: *Concepts in Thermal Physics* (Oxford University Press)**

Waldram JR: *The theory of thermodynamics *(Cambridge University Press)**

*Solid State*

-Hook JR and Hall HE; Solid State Physics (Wiley)***

Kittel C; Introduction to solid state physics (Wiley) ***